Similar to the Sine and Cosine function, we will look at the Unit circle to understand where the Tangent function comes from. Recall that for any point \(P(x,y)\) on the unit circle and the corresponding terminal angle \(\theta\), that \(\tan(\theta)=\frac{{y}}{{x}}\). Since the tangent function involves a fraction, we must consider domain restrictions, namely, when \(x=0\). Hence, whenever we approach \(x=0\), we will see that the tangent function has asymptotic behavior.

From this, we can now determine some of the key features of the Tangent function. First of all, the Period of \(\tan(x)\) is just \(\pi\). I find this easiest to visualize by looking at one period near the origin, which is the horizontal/vertical intercept but also demonstrates most clearly one full period from \(-\pi/2\) to \(\pi/2\). Since the cosine is zero at those values, we find asymptotic behavior there. More generally, given

\[ f(x) = A\tan(B(x-\omega)) \]

The period of \(f\) is given by \(P = \dfrac{\pi}{{B}}\), there is a phase shift (horizontal transformation) by \(\omega\) and \(A\) determines if there is a reflection over the \(x\) axis (notably there is not an analogous "amplitude" for Tangents). We could also include the vertical shift, but it is uncommon in most scenarios involving tangent function.

To graph a Tangent function, determine the phase shift and period. Then, sketch the asymptotes at the half period after and before the phase shift. At the phase shift is a horizontal intercept and then the sign of A determines whether the tangent goes down or goes up. Here is a little Desmos example: